Question

the given diagram shows the parts of a right triangle with an altitude to the hypotenuse. using the two given measures, find the other four.
( s _ { 1 } = 10, u _ { 2 } = 2 sqrt { 14 } )
( s _ { 2 } = square quad h = square quad u _ { 1 } = square quad a = square )
(simplify your answers. type exact answers, using radicals as needed.)

Explanation:

Step1: Find \(s_{2}\) using geometric mean theorem

In a right - triangle with an altitude to the hypotenuse, \(s_{1}^{2}=s_{2}\times h\) and also \(u_{2}^{2}=s_{2}\times h\) (by the geometric mean relationships). But we can use the fact that \(s_{1}^{2}=s_{2}\times h\) and \(u_{2}^{2}=s_{2}\times h\) is not the right approach. The correct formula is \(u_{2}^{2}=s_{2}\times h\) and \(s_{1}^{2}=s_{2}\times h\) is wrong. The right formula for the segments of the hypotenuse: \(a^{2}=s_{1}\times s_{2}\), \(u_{1}^{2}=s_{1}\times h\), \(u_{2}^{2}=s_{2}\times h\), and \(h = s_{1}+s_{2}\). Also, from the geometric mean theorem \(u_{1}^{2}=s_{1}\times h\) and \(u_{2}^{2}=s_{2}\times h\), and \(a^{2}=s_{1}\times s_{2}\). Another way: Using the proportion \(\frac{s_{1}}{a}=\frac{a}{s_{2}}\) (similar triangles). But first, use \(u_{2}^{2}=s_{2}\times h\) and \(h=s_{1} + s_{2}\). Wait, no. The formula \(u_{2}^{2}=s_{2}\times h\) is wrong. The correct formula is \(u_{2}^{2}=s_{2}\times(s_{1}+s_{2})\). Let \(x = s_{2}\), then \((2\sqrt{14})^{2}=x(x + 10)\).

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Since \(x>0\), \(s_{2}=\frac{- 10 + 18}{2}=4\).

Step2: Find \(h\)

Since \(h=s_{1}+s_{2}\), and \(s_{1} = 10\), \(s_{2}=4\), then \(h=10 + 4=14\).

Step3: Find \(u_{1}\)

Using \(u_{1}^{2}=s_{1}\times h\) (geometric mean theorem: in a right - triangle with altitude \(a\) to the hypotenuse \(h\), the leg \(u_{1}\) satisfies \(u_{1}^{2}=s_{1}\times h\)). Substitute \(s_{1}=10\) and \(h = 14\), then \(u_{1}=\sqrt{10\times14}=\sqrt{140}=2\sqrt{35}\).

Step4: Find \(a\)

Using \(a^{2}=s_{1}\times s_{2}\) (geometric mean theorem: \(a\) is the geometric mean of \(s_{1}\) and \(s_{2}\)). Substitute \(s_{1}=10\) and \(s_{2}=4\), then \(a=\sqrt{10\times4}=2\sqrt{10}\).

Answer:

\(s_{2}=4\), \(h = 14\), \(u_{1}=2\sqrt{35}\), \(a=2\sqrt{10}\)